This essay will explore the relationship between arithmetic and geometric means. First, we will prove that the arithmetic means is always greater than or equal to the geometric means by an algebraic proof. Then, we will visually display and demonstrate this relationship by spreadsheet using Microsoft Excel and two sketches using "Geometric Sketch Pad, Enhanced Version 3.00".

Given two values **X **and **Y**, both being greater than or equal to
zero, we will prove that the arithmetic mean (
)
is greater than or equal to the geometric mean (
).
We know that the following is true:

A comparison of arithmetic mean and geometric mean of a set a values for
**a **and **b **can be easily visualized in a spreadsheet. The following
is an example of a couple of rows from the attached Excel spreadsheet depicting
values for **a **and **b **stored in columns **A **and **B**,
respectively. The arithmetic mean is calculated in column **C**, with the
geometric mean stored in column **D**. Example:

A B C D 1. 1 10 5.5 3.16227766 2. 5 99 52 22.2485955 3. 20 20 20 20

**Spreadsheet Attachment** (Select to load the Excel Spreasheet with more data .).

Through use of Geometry Sketchpad, two segments of length **A **and **B**
can be used to demonstrate that the arithmetic mean is always greater than the
geometric mean except when **A **and **B **are equal (both means are
equal under this condition) . The following sketch shows the two segments with
and a circle created with a diameter of the combined segments. The radius
**CD **represents the arithmetic mean
.
By dropping a perpendicular segment from the point **D** on the circle to
point **E **on the line **m**, we get a right trianlge with side
**CE** having a length of
.
By knowing the length of the hypotenuse **CD **and one of the sides
**CE**, we can calculate the length of the other side **DE **as
follows:

When segment B and segment A are equal in length, the following sketch shows that arithmetic and geometric mean are equal.

**GSP Diameter Demonstration**.If you have GSP Enhanced, Verison 3, this will load the above sketch with animation that will demonstrate the full range of values for segments A and B.

By taking those same two segments, A and B, we can create a sqare with sides of the combined lengths of A and B. The total area of the square will be while the volume of the four rectangles with sides A and B are . The former volume will always be greater than or at best, equal to the latter as shown algebraically as follows:

Once again, we have the left side of the equation representing the aritmetic mean and the right side representing the geometric mean. Please launch the following sketch (by selecting) to see how the square is set up.

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